Mathematical Rings
Author
YOGESH MALIK
Last Updated
5년 전
License
Creative Commons CC BY 4.0
Abstract
Mathematical Rings
Mathematical Rings
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\title{RING}
\subtitle{\Large{Delhi Technological University \\ DELHI}}
\author{Submitted By : YOGESH MALIK }
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\frametitle{RING}
\textcolor{blue}{
\textcolor{black}{ $\bullet$ $DEFINITION:-$}
\\ A non-empty set R , equipped with two binary operations called addition and multiplication denoted by (+) and (.) is said to form a ring if the following properties are satisfied :\textcolor{black}{\\Properties under Addition :} \\ $1$. $R$ is closed with respect to addition \\i.e., $a,$ $b$ $\in$ $R$, then $a + b$ $\in$ $R$\\ $2$. Addition is associative \\i.e., $a + (b + c) = (a + b) + c$ $\forall$ $a, b, c$ $\in$ $R$\\ $3$. Addition is commutative \\i.e., $a + b=b + a$ $\forall$ $a$, $b$ $\in$ $R$}
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\textcolor{blue}{$4$. Existence of additive identity\\i.e., there exist an additive identity in R denoted by in R such that \\$0+a=a=a+0$ $\forall$ $a$ $\in$ $R$\\$5$. Existence of additive inverse
\\i.e., to each element $a$ in $R$, there exists an element $–a$ in $R$ such that \\ $-a + a = 0 = a + (-a)$\textcolor{black}{\\Properties under Multiplication :} \\ $6$. $R$ is closed with respect to multiplication \\ i.e., if $a, b \in$ $R$, then $a .b$ $\in$ $R$ \\ $7$. Multiplication is associative \\ i.e., $a. (b .c) = (a. b).c$ $\forall$ $a, b, c$ $\in$ $R$\\ $8$. Multiplication is distributive with respect to addition \\ i.e., $\forall$ $a, b, c$ $\in$ $R$ , $a. (b + c) = a. b + a .c$ [Left distributive law] \\ And $(b + c) . a = b. a + c. a $[Right distributive law]}
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\textcolor{blue}{$\bullet$ REMARK:
\hspace{4cm}\\Any algebraic structure ($R$, $+$, $.$) is called a ring if ($R$, $+$) is an abelian group and $R$ is closed , associative with respect to multiplication and multiplication is distributive with respect to addition.}
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\begin{frame}{$\circledast$ TYPES OF RING }
\textcolor{black}{$1$. COMMUTATIVE RING :}\textcolor{blue}{ \\A ring in which $a. b = b .a$ $\forall$ $a, b$ $\in$ $R$ is called commutative ring.}\textcolor{black}{\\ 2. RING WITH UNITY :}\textcolor{blue}{\\If in a ring, there exist an element denoted by $1$ such that $1.a=a=a.1$ $\forall$ $a$ $\in$ $R$ is called a ring with unity element.\\ The element $1$ $\in$ $R$ is called the unit element of the ring. \\ Thus, if $R$ satisfies the all eight properties of ring and also have multiplicative identity, then we define $R$ as ring with identity. }
\textcolor{black}{\\$3$. NULL RING OR ZERO RING :} \textcolor{blue}{\\The set $R$ consisting of a single element $0$ with two binary operations defined by $0+0=0$ is a ring and is called null ring or zero ring.}
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\textcolor{red}{ \vspace{2cm}\\ Eg. Prove that the set $Z$ of all integers is a ring with respect to addition and multiplication of integers.}
\textcolor{black}{\\Proof:
\\ $\centerdot$ Properties under Addition :}
\textcolor{blue}{\\$1$. Closure property: As sum of two integers is also an integer ,\\ ∴ $Z$ is closed with respect to addition of integers .
\\$2$. Associativity: As addition of integers is also an associative composition
\\ $\therefore$ , $a + (b + c) =(a + b) +c$ $\forall$ $a , b , c$ $\in$ $Z$
\\$3$. Existence of additive identity: For $0$ $\in$ $Z$, $0 + a = a = a +0$ $\forall$ a $\in$ $Z$.
\\$\therefore$, $0$ is additive identity.
\\$4$. Existence of additive inverse: For each $a$ $\in$ $Z$ there exist $–a$ $\in$ $Z$ such that $a + (-a) = 0 = (-a) + a$ ,
\\where $0$ is identity element .}
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\vspace{1cm}\textcolor{blue}{$5$. Commutative property :
\\ $a + b = b+ a$ $\forall$ $a , b$ $\in$ $Z$\\ } \textcolor{black}{ $\centerdot$Properties under Multiplication:}\textcolor{blue}{\\6. Closure property with respect to multiplication: As product of two integers is also an integer\\$a . b$ $\in$ $Z$ $\forall$ $a, b$ $\in$ $Z$\\7. Multiplication is associative :\\ $a . (b .c) = (a .b) .c$ $\forall$ $a, b, c$ $\in$ $Z$ \\ 8. Multiplication is distributive with respect to addition:\\$\forall$ $a, b, c$ $\in$ $Z$, $a. (b + c) = a .b + a .c$ \\And $(b + c) .a = b .a + c .a$\\Hence,$Z$ is a ring with respect to addition and multiplication of integers.}
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\textcolor{black}{$\blacktriangleright$ Note:}\textcolor{blue}{\\
1. As $1 .a = a .1 = a$ , $\forall$ $a$ $\in$ $Z$ ,\\ $\therefore$ $1$ is a multiplicative identity of $Z$.
\\2. As $a .b = b .a$, $\forall$ $a , b$ $\in$ $Z$,
\\ $\therefore$ multiplication of integers is commutative .
\\ Hence, $Z$ is a commutative ring with unity.}\textcolor{black}{
\\$\circledast$ $Remark:$}\textcolor{blue}{
\\ A ring $R$ is said to be Boolean ring if $x^2 = x$ $\forall$ $x \in R$.}
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\textcolor{black}{\\ Eg. Prove that a ring $R$ in which $x^2 = x$ $\forall$ $x \in R$ , must be commutative.
\\ OR
\\ Show that a Boolean ring is commutative.
}\textcolor{blue}{\\
Proof:
\\ Let $x, y \in R$ $\Rightarrow$
$x + y \in R$
\\ By give condition, $(x + y) ^2 = x + y$ $\forall$ $x, y \in R$
\\ $\Rightarrow$ $(x + y)(x + y) = x + y $
\\ $\Rightarrow$ $x. x + x. y + y. x + y. y = x + y$
\\ $x^2 + x. y + y. x + y^2 = x + y$
\\$\Rightarrow$ $x + x. y + y. x + y = x + y$ [$\therefore$ $x^2 = x$ , $y^2 = y$ ]
\\ $\Rightarrow$ $x. y + y. x = 0$
\\$\Rightarrow$ $x. y = -(y .x)$
\\ $x.y = ( -y .x)^2$ ………(1)}
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Again $\forall$ $y \in R$ , $(y + y) ^2 = y + y$
\\$\Rightarrow$ $(y + y)(y + y) = y + y$
\\$\Rightarrow$ $y. y + y .y + y .y + y .y = y + y$
\\ $y^2 + y^2 + y^2 + y^2 = y + y$
\\$\Rightarrow$ $y + y + y + y = y + y$
\\$\Rightarrow$ $y + y = 0$
\\$\Rightarrow$ $y = -y$ \\
$\therefore$ from (1), $x. y = (yx)^2$
\\ $x.y = yx$
\\Thus $x .y = y .x$
$\forall$ $x, y \in R$
\\Hence,$R$ must be commutative. }
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\begin{frame}{$\circledast$ RINGS WITH OR WITHOUT ZERO DIVISORS:}
\textcolor{blue}{\\A ring $(R, + , .)$ is said to be $without$ $zero$ $divisors$ if for all $a$, $b$ belong to R $a. b = 0$ that implies either $a = 0$ or $b = 0$ \\On the other hand, if in a ring $R$ there exists non zero elements $a$ and $b$ such that $a. b =0$, then $R$ is said to be a $ring$ $with$ $zero$ $divisors.$ \\Eg. \\1. Sets $Z$, $R$, $C$, and $Q$ are without zero divisors rings.\\2. The ring (${0, 1, 2, 3, 4, 5}$, $+$6, $×$6) is a ring with zero divisors.}
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{
Eg. Prove that the set $\{0,1,2,3,4,5 \}$ with addition modulo $6$ and multiplication modulo $6$ as composition is a ring with zero divisors.}
\textcolor{blue}{
Proof :
\\ Let $R$ =$\{0,1,2,3,4,5\}$}
\textcolor{black}{ \\Properties under addition :}
\textcolor{blue}{
\\1. Closure law :
\\As all the entries in the addition composition table are elements of set $R$ is closed w.r.t. addition modulo 6.
\\2. Associative law : \\The composition $+6$ is associative. If $a,b,c$ are any three elements of $R$ then
\\ $a$ +6 $(b$ +6 $c)$= $a$ +6 $(b + c)$
\\ $a$ +6 ($b$ +6 $c$)= least non-negative remainder when $a+(b+c)$ is divided by $6$}
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\\$a +6 (b+6 c)$=least non-negative remainder when $(a+b)+c$ is divided by $6$
\\$a$ +6 ($b$ +6 $c$)=$(a+b)$ +6 $c$
\\$a$ +6 ($b$ +6 $c$)=($a$ +6 $b$) +6 $c$
\\3. Existence of identity :
\\As $0$ +6 $a$ = $a$=$a$ +6 $0$ $\forall$ $a$ $\in$ $R$
\\4. Existence of inverse :
\\From the table , we see that the inverse of $\{0,1,2,3,4,5\}$ are $\{0,5,4,3,2,1\}$ respectively. Hence , additive inverse exists.
\\5. Commutative law :
\\ For all $a,b$ $\in$ $R$ , we have $a$ +6 $b$=$b$+6 $a$}
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\textcolor{black}{
\\Properties under multiplication :}
\textcolor{blue}{
\\6. Closure law for multiplication : \\All the entries in the multiplication composition table are element of set $R$ , therefore $R$ is closed with respect to multiplication modulo 6.}
\textcolor{blue}{\\7. Associative law for multiplication : \\Let $a, b , c$ $\in$ $R$
\\ $\therefore$ $a$ ×6 ($b$ ×6 $c$) = $a$ ×6 $(b c)$
\\ $a$ ×6 $(b$ ×6 $c)$ = least non – negative remainder when $a(b c)$ is divided by 6.
\\ $a$ ×6 $(b$ ×6 $c)4$ = least non negative remainder when $(ab)c$ is divided by 6
\\ $a$ ×6 $(b$ ×6 $c)$ = $ab$ ×6 $c$
\\ $a$ ×6 $(b$ ×6 $c)$ = $(a$ ×6 $b)$ ×6 $c$}
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\textcolor{blue}{
8. Distribution laws : \\If $a,b,c$ be any three elements of $R$ , then
\\$a$ ×6 $(b$ +6 $c)$ = $a$ ×6 $(b$ + $c$)
\\$a$ ×6 $(b$ +6 $c)$= least non negative remainder when $a(b+c)$ is divided by 6
\\$a$ ×6 $(b$ +6 $c)$ = least non – negative remainder when $ab+ac$ is divided by 6
\\$a$ ×6 $(b$ +6 $c)$ = $ab$ +6 $ac)$
\\$a$ ×6 $(b$ +6 $c)$ = $a$ ×6 $(b$ +6 $c)$}
\textcolor{blue}{
\\similarly , ($b$ +6 $c)$ ×6 $a$ = $(b$ ×6 $a)$ +6 $(c$ ×6 $a)$
\\Hence , $R$ is a ring with respect to given compositions.}
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\textcolor{blue}{\\As $(R$, $+$6, $×$6) is ring ,
\\Now for $2, 3$ \in R , $2× 3= 0$
\\i.e., product of two non zero element is equal to the zero element of the ring .\\
Hence , $R$ is a ring with zero divisors.}
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